Actively stabilized random number generator

ABSTRACT

An actively stabilized random number generator includes a random number generator and a feedback controller. The random number generator includes a chaotic physical circuit realizing an iterated function. The iterated function is configured to produce a trajectory of iterates and has an operating parameter β and a desired Markov operating point. A binary bit converter has a symbol function configured to produce binary symbols from the trajectory of iterates and a maximal kneading sequence. The feedback controller is configured to observe the maximal kneading sequence within the trajectory of iterates and adjust the operating parameter to the desired Markov operating point.

FIELD

The present disclosure generally relates to cryptography, and morespecifically, to random number generation via chaotic physical systems.

BACKGROUND

Random numbers are used in a multitude of fields such as encryption,computer simulations, artificial scene generation, and gambling. Mostdevices that are used to generate random numbers are pseudo-randomnumber generators (PRNGS). However, PRNGS are not true-random becausethey are based on systems and/or algorithms with inherent, and thusdeterminable, patterns. For applications such as encryption, true-randomgeneration is important to securing communication. Otherwise, deviationfrom true-random generation creates potential for said encryption to be“broken” (i.e., the random number generator possesses exploitablevulnerability). For example, private communication, data sharing, andcommerce are just a few applications where unbreakable encryption isbecoming increasingly important.

Currently, devices generating true-random numbers often rely on specialequipment based on quantum uncertainty or complex chaotic classicaldynamics. However, devices based on quantum uncertainty are costly,slow, and are therefore not suitable for large-scale integration, anddevices based on complex chaotic classical dynamics defy completeanalysis and can exhibit vulnerabilities due to non-ideal realization.All generators based on physical properties are sensitive to changes inenvironmental parameters, such as ambient temperature.

Therefore, there is a need for true-random generators withoutvulnerabilities that have the ability for large-scale integration andare tolerant to environmental change.

BRIEF SUMMARY

An actively stabilized random number generator includes a random numbergenerator and a feedback controller. The random number generatorincludes a chaotic physical circuit designed to realize an iteratedfunction. The iterated function of the random number generator has asingle input state on an interval. The iterated function furtherincludes a single output state being on the same interval. The iteratedfunction is configured to have a trajectory of iterates, wherein eachiterate is within the interval. The iterated function having a positiveentropy rate, wherein the entropy rate defines a rate at which randominformation is produced. The iterated function defines a unimodal map,with the unimodal map having a slope configuration. The iteratedfunction has an operating parameter. The operating parameter is on theinterval 1<β≤2. The operating parameter corresponds with the slopeconfiguration of the unimodal map. The operating parameter correspondingto the entropy rate of the iterated function. The operating parameterhas a desired Markov operating point.

BRIEF DESCRIPTION OF DRAWINGS

For a better understanding of the nature and objects of the disclosure,reference should be made to the following detailed description taken inconjunction with the accompanying drawings, in which:

FIG. 1 is a diagram of a chaotic circuit realizing a unimodal map and afeedback controller.

FIG. 2 is a diagram of a unimodal map.

FIG. 3 is a flow chart showing functions performed by the activelystabilized random number generator.

Reference is made in the following detailed description of preferredembodiments to accompanying drawings, which form a part hereof, whereinlike numerals may designate like parts throughout that are correspondingand/or analogous. It will be appreciated that the figures have notnecessarily been drawn to scale, such as for simplicity and/or clarityof illustration. For example, dimensions of some aspects may beexaggerated relative to others. Further, it is to be understood thatother embodiments may be utilized. Furthermore, structural and/or otherchanges may be made without departing from claimed subject matter.References throughout this specification to “claimed subject matter”refer to subject matter intended to be covered by one or more claims, orany portion thereof, and are not necessarily intended to refer to acomplete claim set, to a particular combination of claim sets (e.g.,method claims, apparatus claims, etc.), or to a particular claim.

DETAILED DESCRIPTION

The present disclosure provides an actively stabilized random numbergenerator (ASRNG) 10, as schematically illustrated in FIG. 1. The ASRNG10 of the present disclosure generates true random outputs forapplications such as, but not limited to, encryption, computersimulations, and compressive sensing. As one skilled in the art wouldknow, “true random” outputs are unpredictable whereas “pseudo-random”outputs are deterministic function outputs with suitable statisticsdesigned to mimic true random numbers. As schematically illustrated inFIG. 1, the actively stabilized random number generator 10 has a randomnumber generator, a binary bit converter 13, and a feedback controller16.

As seen in FIGS. 1 and 2, the random number generator of the ASRNG 10has a chaotic physical circuit 18, wherein the chaotic physical circuit18 is constructed to have an iterated function. The iterated functiondefines a unimodal map 22, the unimodal map being a tent map, and theiterated function has a single input state, a single output state, atrajectory of iterates 28, a positive entropy rate, and an operatingparameter.

As seen in FIG. 2, the unimodal map 22 of the iterated function has areduced-slope configuration 30. As one of ordinary skill in the artwould know, a unimodal map 22 may have a full-slope configuration 31,but for the purpose of this disclosure the reduced-slope configuration30 is realized because the full-slope configuration is structurallyunstable. The operating parameter is denoted using the symbol “β”. Theoperating parameter corresponds with the full-slope configuration 31 andthe reduced-slope configuration 30 of the unimodal map 22 and is on theinterval 1<β<2. The operating parameter also has a desired Markovoperating point. The desired Markov operating point being defined as acorresponding value at which the iterates are considered Markov, orfinitely dependent.

In the equation below, the single input state is denoted by the variable“x_(n)”, with the subscript “n” denoting an iteration number. Generally,the iteration number represents time. The iterated function isconfigured to input the single input state and output the single outputstate. The single input state is on an interval 0<x_(n)<1. Each of thesingle output states create an iterate 32 of the trajectory of iterates28 of the unimodal map 22, as seen in FIG. 2. The trajectory of iterates28 are within the same interval of the single input state.

$x_{n + 1} = {\beta \cdot \{ \begin{matrix}{x_{n},} & {x_{n} \leq {1/\beta}} \\{{1 - x_{n}},} & {x_{n} > {1/\beta}}\end{matrix} }$

As one skilled in the art would know, a different trajectory of iterates28 can be produced given different initial input states. For example,given two scenarios, the first scenario initial input state has a smalldifference between an initial input of the second scenario, eventuallyresulting in vastly differing trajectories. The iterated functionresults in the trajectory of iterates 28 as the iteration number isincreased, wherein the trajectory of iterates 28 define an overall shapeof the unimodal map 22, as shown in FIG. 2.

As seen below by the following equation, a rate at which randominformation is produced by each trajectory of iterates 28 defines thepositive entropy rate, denoted by “h”, which has units of bits periteration, and is calculated using the operating parameter:

h=log ₂(β)

In the present disclosure, the value of the operating parameter of theactively stabilized random number generator (ASRNG) 10 is β<2,corresponding with the positive entropy rate being h<1 bit per iterationby the equation above.

Each one of the trajectory of iterates 28 are then processed by thebinary bit converter 13 of the ASRNG 10. The binary bit converter 13 isconfigured to have a symbol function defined by the following equation:

$S_{n} = \{ \begin{matrix}{0,} & {x_{n} \leq {1/\beta}} \\{1,} & {x_{n} > {1/\beta}}\end{matrix} $

The symbol function produces a binary symbol, which is a bit, for eachiterate 32 of the trajectory of iterates 28, denoted in the equationabove by “S_(n)”. Generally, a sequence of bits is produced as thesymbol function translates a plurality of iterates of the iteratedfunction as the iterated function increases in iteration number. Anatural partition, relative to a critical point 33 of the unimodal map22, the critical point 33 being configured by the operating parameter,is utilized within the symbol function, resulting in the symbol functionhaving a symbol rate, which is one bit per iteration. The symbol ratebeing defined as the rate at which bits are produced via the symbolfunction. As one skilled in the art would know, the entropy rate cannotexceed the symbol rate, and the amount by which the symbol rate exceedsthe entropy rate is due to bias and interdependence in binary symbolsproduced by the binary bit converter 13. Methods to remove bias from thebinary symbols are known to one skilled in the art. Interdependence,which is defined as a dependence on prior outcomes of the iteratedfunction, results in vulnerability within the ASRNG 10. Thevulnerability is mitigated using the feedback controller 16, asdescribed in further detail below.

To be able to generate independent random binary symbols, the feedbackcontroller 16 is integrated into the chaotic physical circuit 18 tocorrect for interdependence within the ASRNG 10, as a result of thepositive entropy rate being h<1 bit per symbol, and to automaticallycorrect for physical deviations due to circuit implementation (e.g.,manufacturing defects or environmental changes). The feedback controller16 exploits the Markov properties associated with the operatingparameter utilizing kneading theory. As one skilled in the art wouldknow, kneading theory states that the desired Markov operating point ofthe operating parameter is defined by a unique maximal kneadingsequence. As a result, each iterate 32 of the iterated function isconsidered Markov and emits random sequences that order less than orequal to its corresponding maximal kneading sequence. The feedbackcontroller 16 observes the maximal kneading sequence within thetrajectory of iterates 28 to determine if the operating parameter needsto be increased or decreased in order to maintain the iterates at thedesired Markov operating point, and to compensate for the physicaldeviations of the chaotic physical circuit 18 that may cause theoperating parameter to change. One skilled in the art would know thatmethods exist for extracting random bit sequences without bias andinterdependence from the system with the desired Markov operating point.

Described above is the mathematical construction of the activelystabilized random number generator 10. As previously mentioned, theactively stabilized random number generator 10 can be realized using thechaotic physical circuit 18, as seen in FIG. 1. As one skilled in theart would know, the chaotic physical circuit 18 contains both analog anddigital circuitry. The analog circuitry contains a capacitor C 34 and anegative resistor −R 36, as seen in FIG. 1. The practical realization ofan electronic negative resistor is known to one skilled in the art. Thedigital circuitry has a RS flip flop 38 and a clocked D flip flop 40. Inaddition, the chaotic physical circuit 18 has a comparator 42 fordetecting if voltage above the capacitor C 34 exceeds a threshold V_(T)44 and a buffer circuit that converts digital to analog voltages where adigital F 46 generates a zero 48, and a digital T 50 generates twice thethreshold 2V_(T) 52. An input clock signal 54 is a periodic impulse of ashort duration that resets an output of the RS flip flop 38 and loadsdata into the clocked D flip flop 40. As one skilled in the art wouldunderstand, a short delay τ 56 is included between the RS flip flop 38and the clocked D flip flop 40 to allow the D flip flop to latch the RSflip flop output prior to the RS flip flop's reset via the clock signal54. As seen in FIG. 1, the chaotic physical circuit 18 outputs a digitalsignal S 58 and the clock signal 54 which are synchronized, relative toa feedback controller portion 60 of the chaotic physical circuit. Thedigital signal S 58 exhibits a new random bit with each iterate 32 ofthe iterated function. Random numbers of the ASRNG system 10 are derivedfrom the unimodal map 22 utilizing the operating parameter. Theoperating parameter value “β” is derived using the following equation:

$\beta = {\exp\{ \frac{T}{RC} \}}$

In the equation above, “exp” indicates an exponential function, “T” is aclock period, “R” is a magnitude of a negative resistance, and “C” is acapacitance. The chaotic circuit 18 is designed such that “T” of theclock period is within a parameter of 0<T<RC ln(2). When “R” is themagnitude of the negative resistance is multiplied by “C” is thecapacitance and the natural logarithm 2; such that the parameters forthe unimodal map 22 are 1<β<2.

FIG. 1 further shows a schematic illustration of the feedback controller16 integrated into the chaotic physical circuit 18, forming the feedbackcontroller portion 60 of the chaotic physical circuit. The outputs ofthe chaotic physical circuit 18 relative to the feedback controller 16are S 58 and the synchronized clock signal 54 signals, which are thenentered into a shift register 62 of the feedback controller portion 60.Random bits that are conveyed on S 58 are converted to a parallelrepresentation using the shift register 62 driven by the clock signal54. A fixed register 64 stores the maximal kneading sequencecorresponding to the desired Markov operating point. A size of the shiftregister 62 in bits is matched to a size of the fixed register 64 thatis required to store the maximal kneading sequence. A parallel output ofthe shift register 62 is bit-wise compared to the maximal kneadingsequence in a XOR/AND block 66 of the feedback controller portion 60. Adigital output from a comparison from the XOR/AND block 66 indicates atrue state, if and only if, contents of the shift register 62 exactlymatch the maximal kneading sequence stored in the fixed register 64.

The digital output of the comparison from the XOR/AND block 66 isconverted to two analog voltage level signals, with a logical Ftranslating to voltage V1 68 and a logical T to voltage V2 70. Theseanalog voltage level signals are the equivalent of an analog signal 72.The analog signal 72 passes through a low-pass filter 74 with a timeconstant that is sufficiently long to yield an output voltage V3 signal76 that is inversely proportional to a relative occurrence of themaximal kneading sequence within a random bit stream. The output voltageV₃ signal 76 is added to a fixed voltage V₄ 78 that provides a coarseoperating point to aid targeting the desired Markov operating pointusing an output signal V_(F) 80. The feedback controller 16 outputsignal V_(F) 80 can be used to control the operating parameter viavoltage control of either the negative resistance −R 36, the capacitanceC 34, or a clock period. For purposes of clarity, FIG. 3 schematicallyillustrates the steps of the ASRNG 10 in accordance with the presentdisclosure.

The present disclosure provides the chaotic physical circuit 18 thatgenerates random numbers and may be built using readily availableelectronic parts. Detection of the maximal kneading sequence within thetrajectory of iterates 28 of the iterated function provides a built-indiagnostic for monitoring the positive entropy of the random numbergenerator, forming the ASRNG 10.

The foregoing description has been presented for the purposes ofillustration and description. It is not intended to be exhaustive or tolimit the disclosure to the precise form disclosed. Many modificationsand variations are possible in view of this disclosure. Indeed, whilecertain features of this disclosure have been shown, described and/orclaimed, it is not intended to be limited to the details above, since itwill be understood that various omissions, modifications, substitutionsand changes in the apparatuses, forms, method, steps and systemillustrated and in its operation can be made by those skilled in the artwithout departing in any way from the spirit of the present disclosure.

Furthermore, the foregoing description, for purposes of explanation,used specific nomenclature to provide a thorough understanding of thedisclosure. However, it will be apparent to one skilled in the art thatthe specific details are not required in order to practice thedisclosure. Thus, the foregoing descriptions of specific embodiments ofthe present disclosure are presented for purposes of illustration anddescription. They are not intended to be exhaustive or to limit thedisclosure to the precise forms disclosed, many modifications andvariations are possible in view of the above teachings. The embodimentswere chosen and described in order to best explain the principles of thedisclosure and its practical applications, to thereby enable othersskilled in the art to best utilize the disclosed system and method, andvarious embodiments with various modifications as are suited to theparticular use contemplated.

What is claimed:
 1. An actively stabilized random number generatorcomprising: a random number generator; and a feedback controller,wherein the random number generator comprises: a chaotic physicalcircuit that realizes an iterated function, the iterated function havinga single input state on an interval and a single output state within theinterval, the iterated function configured to have a trajectory ofiterates, wherein each iterate is within the interval, the iteratedfunction having a positive entropy rate that defines a rate at whichrandom information is produced; wherein the iterated function defines aunimodal map, the iterated function having an operating parameter (β) onan interval of 1<β≤2, the operating parameter determining a slopeconfiguration of the unimodal map, the operating parameter correspondingto the entropy rate of the iterated function, the operating parameterhaving a desired Markov operating point; and wherein the feedbackcontroller is configured to observe a maximal kneading sequence withinthe trajectory of iterates and to adjust the operating parameter to thedesired Markov operating point.
 2. The actively stabilized random numbergenerator of claim 1, wherein the feedback controller is furtherconfigured to maintain the operating parameter at the desired Markovoperating point.
 3. The actively stabilized random number generator ofclaim 1, wherein the actively stabilized random number generator furthercomprises a binary bit converter, the binary bit converter realizing asymbol function, the symbol function having a natural partition and asymbol rate, the natural partition being configured to generate binarysymbols, the symbol rate being one binary symbol generated periteration, the symbol rate being greater than or equal to the entropyrate.
 4. The actively stabilized random number generator of claim 1,wherein the slope configuration of the unimodal map is a reduced slopeconfiguration.
 5. The actively stabilized random number generator ofclaim 4, wherein the symbol rate is less than the entropy rate of thechaotic physical circuit.
 6. The actively stabilized random numbergenerator of claim 4, wherein the reduced slope configuration has thecondition β<2.
 7. The actively stabilized random number generator ofclaim 1, wherein the slope configuration of the unimodal map is a fullheight configuration.
 8. The actively stabilized random number generatorof claim 7, wherein the full slope configuration has the condition β=2.